VOTO PARTICULAR
JESÚS ERNESTO PECES MORATE.
The equations (1.9) to (1.11) require additional assumptions to determine the steady fluxes of T and S. In the following we discuss the way in which salt finger fluxes scale with the S gradients and differences between the fluxes through thin interfaces and deep linear gradients.
1.2.3.1 Thin interfaces
Most of the previous laboratory experiments on salt fingers have concentrated on the case where there is a well-mixed layer of hot salty water overlying colder, fresher water. Salt fingers can form at the interface between the two layers when 1 α∆T
β∆S τ 1, whereα∆T and
β∆S are the respective T and S difference between the two well-mixed reservoirs, andαandβ are the T and S expansion co-efficients.
For interfaces the dimensionless T flux, Nu, must be a function of the following variables
Nu κ FT T∆T
d
fgβ∆S gα∆T d ν κS κT (1.14)
where d is the convective layer depth, g the acceleration due to gravity,κSandκT the respective molecular diffusivities of S and T , andνis the viscosity of the fluid. These variables can be formed into of the non-dimensional parameters
Pr κν T τ κS κT Ra gα∆T d 3 νκT RaS gβ∆Sd3 νκT (1.15)
giving Nu
fRa RaS Prτ . Turner (1967) assumed that the buoyancy flux only depended upon the properties of a thin boundary layer at the edge of the convecting region and did not depend upon the convective layer depth d or the Prandtl number, Pr. It then follows from the definition of Ra and Nu that
Nu f β∆S
α∆T τ
Ra1 3 (1.16)
since this is the only form that removes the dependence of FT on d. Using the definition of the Nusselt number (1.14), and the Rayleigh number (1.15), Turner (1967) then showed that
βFT fRρ τ β∆S
4 3 (1.17)
The value of fRρ τ was measured by Stern & Turner (1969) as f Rρ τ 10
2cm s 1in
sugar/salt experiments. Similar experiments by Lambert & Demenkow (1972) at lower values of Rρ, found values of fRρ τ in the range of 0 5 10
3to 0 75 10 3cm s 1. Part of the
reason for the difference between these results was explained by Griffiths & Ruddick (1980) who measured the dependence of fRρ τ upon Rρ and found that fRρ τ
Rρ6. Recent numerical simulations of the salt finger fluxes across thin interfaces by Radko & Stern (2000) have also verified the functional form of (1.17).
1.2.3.2 Deep linear gradients
An interface between two well-mixed regions is no longer ‘thin’ when the thickness is much larger than the intrinsic finger length scale and the fingers are no longer connected to both of the well-mixed reservoirs. This occurs as salt fingers are not infinitely long (as assumed in 1.8) but rather have aspect ratio ratio of two or three, for Rρ 2 (Taylor, 1993; Shen, 1995). For salt fingers in deep ‘crossed’ gradients of S and T , the result of (1.17) does not apply. The extensive parameters which govern the dynamics of the salt fingers in deep linear gradients are gβSz gαTz ν κS κT, and there is no externally imposed length scale. These variables can be expressed in terms of the non-dimensional parameters
Pr ν κT τ κS κT Rρ αTz βSz (1.18)
1.2 The interaction between salt fingers, shears and intermittent turbulence 13
and there is no Rayleigh number. The Nusselt number in this case is defined in terms of the gradients as
Nu FT
κTTz
f1Rρ Prτ (1.19)
For a given fluid, Pr andτwill be constant. Using the definition (1.19) leads to
FT f2 Rρ κT
∂T
∂z (1.20)
and for a given Rρthe flux of T is linearly proportional to the gradient of T .
The equation (1.20) does not give the magnitude of the T fluxes or their dependence upon Rρ. Various theoretical mechanisms have been proposed by which salt finger fluxes in deep gradients are limited. The initial growth of fingers is exponential, with growth rate (1.12), up to the point where shear instability of the fingers (described by a Froude or Reynolds number criterion) sets in (Kunze, 1987) and steady fluxes are then reached. Holyer (1984,1985) and Taylor & Veronis (1986) have identified vertical disturbances that grow at a wavelength com- parable with the finger width, consistent with observations of Taylor (1993) that salt fingers with 1 Rρ 5 broke down into ‘blobs’ of aspect ratio of two. Direct numerical simulations of salt fingers in deep linear gradients have been used to parameterize the magnitude of the fluxes. In 2D and 3D numerical experiments of salt fingers in linear T and S gradients, Shen (1995), Radko & Stern (1998,2000), Stern & Radko (1999) and Merryfield (2001) developed scaling for the resulting non-dimensional T and S fluxes in terms of Rρ.